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Sep 20, 2013
09/13

by
Jozef H. Przytycki

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Symmetry of geometrical figures is reflected in regularities of their algebraic invariants. Algebraic regularities are often preserved when the geometrical figure is topologically deformed. The most natural, intuitively simple but mathematically complicated, topological objects are Knots. We present in this papers several examples, both old and new, of regularity of algebraic invariants of knots. Our main invariants are the Jones polynomial (1984) and its generalizations. In the first section,...

Source: http://arxiv.org/abs/math/0405151v1

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Sep 18, 2013
09/13

by
Jim Hoste; Laura Zirbel

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Knots in Euclidean space which may be parameterized by a single cosine function in each coordinate are called Lissajous knots. We show that twist knots are Lissajous knots if and only if their Arf invariants are zero. We further prove that all 2-bridge knots and all (3,q)-torus knots have Lissajous projections.

Source: http://arxiv.org/abs/math/0605632v1

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Sep 18, 2013
09/13

by
William W Menasco

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A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transversal knot in the standard contact structure for S^3 is defined to be transversally simple if it is characterized up to transversal isotopy by its topological knot type and its...

Source: http://arxiv.org/abs/math/0002110v2

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Sep 20, 2013
09/13

by
Alan Durfee; Donal O'Shea

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A polynomial knot is a smooth embedding $\kappa: \real \to \real^n$ whose components are polynomials. The case $n = 3$ is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological knot. This paper contains basic results for these knots as well as many examples.

Source: http://arxiv.org/abs/math/0612803v1

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Sep 22, 2013
09/13

by
Colin Adams

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In 1978, W. Thurston revolutionized low diemsional topology with his work on hyperbolic 3-manifolds. In this paper, we discuss what is currently known about knots in the 3-sphere with hyperbolic complements. Then focus is on geometric invariants coming out of the hyperbolic structures. This is one of a collection of articles to appear in the Handbook of Knot Theory.

Source: http://arxiv.org/abs/math/0309466v1

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Sep 18, 2013
09/13

by
Peter Ozsvath; Zoltan Szabo; Dylan Thurston

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Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

Source: http://arxiv.org/abs/math/0611841v2

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Sep 18, 2013
09/13

by
Charles Livingston

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In answer to a question of Long, Flapan constructed an example of a prime strongly positive amphicheiral knot that is not slice. Long had proved that all such knots are algebraically slice. Here we show that the concordance group of algebraically slice knots contains an infinitely generated free subgroup that is generated by prime strongly positive amphicheiral knots. A simple corollary of this result is the existence of positive amphicheiral knots that are not of order two in concordance.

Source: http://arxiv.org/abs/math/0008166v2

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Sep 20, 2013
09/13

by
Tomasz S Mrowka; Yann Rollin

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We prove a generalization of Bennequin's inequality for Legendrian knots in a 3-dimensional contact manifold (Y,xi), under the assumption that Y is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka [Invent. Math. 130 (1997) 209-255] is nonvanishing. The proof requires an excision result for Seiberg-Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying...

Source: http://arxiv.org/abs/math/0410559v6

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Sep 18, 2013
09/13

by
John B. Etnyre; Ko Honda

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We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.

Source: http://arxiv.org/abs/math/0006112v1

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Sep 23, 2013
09/13

by
Allen Hatcher

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We consider the space of all smooth knots in the 3-sphere isotopic to a given knot, with the aim of finding a small subspace onto which this large space deformation retracts. For torus knots and many hyperbolic knots we show the subspace can be taken to be the orbit of a single maximally symmetric placement of the knot under the action of SO(4) by rotations of the ambient 3-sphere. This would hold for all hyperbolic knots if it were known that there are no exotic free actions of a finite cyclic...

Source: http://arxiv.org/abs/math/9909095v1

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Sep 19, 2013
09/13

by
A. Stoimenow

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We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for weak genus two knots. We also study the values of the link polynomials at roots of unity, extending denseness results of Jones. Using these values, examples of knots...

Source: http://arxiv.org/abs/math/0303012v1

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Sep 18, 2013
09/13

by
Seiichi Kamada

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The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.

Source: http://arxiv.org/abs/math/0008092v1

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Sep 18, 2013
09/13

by
Greg Friedman

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We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of Levine on cobordism of sphere knots, we define disk...

Source: http://arxiv.org/abs/math/0401163v1

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Sep 19, 2013
09/13

by
William W. Menasco

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In math.GT/0002110 the author's Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. This result is in error and this erratum pin points the error. In "An addendum on iterated torus knots" a more subtle result is proven resulting in giving a geometric realization of the Honda-Etnyre transverse (2,3)-cable of the (2,3)-torus knot example--Appendix joint with H. Matsuda. (See math.SG/0306330 and math.GT/0610566.)

Source: http://arxiv.org/abs/math/0610565v1

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Sep 18, 2013
09/13

by
John B. Etnyre

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This is a survey paper on Legendrian and transversal knots for Handbook of Knot Theory.

Source: http://arxiv.org/abs/math/0306256v2

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Jul 20, 2013
07/13

by
John B. Etnyre

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We classify positive transversal torus knots in tight contact structures up to transversal isotopy.

Source: http://arxiv.org/abs/math/9906195v2

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Sep 18, 2013
09/13

by
Louis Kauffman; Vassily Olegovich Manturov

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This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a connected sum of two virtual knots $K_{1}$ and $K_{2}$ is trivial, then so are both $K_{1}$ and $K_{2}$. We establish an algorithm, using Haken-Matveev technique, for recognizing virtual knots. This paper may be read as both an introduction and as a research...

Source: http://arxiv.org/abs/math/0502014v3

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Sep 22, 2013
09/13

by
Alessia Cattabriga; Michele Mulazzani

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We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus. The second representation is parametric: every (1,1)-knot can be represented by a 4-tuple of integer parameters. The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly...

Source: http://arxiv.org/abs/math/0501234v2

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Sep 19, 2013
09/13

by
Yuri Chekanov

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We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing Legendrian knots that have the same classical invariants but are not Legendrian isotopic.

Source: http://arxiv.org/abs/math/0304294v1

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Sep 19, 2013
09/13

by
David Hrencecin; Louis H. Kauffman

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This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding its structure in terms of compositions of morphisms, where elementary morphisms are associated to oriented classical and virtual crossings in the diagram.

Source: http://arxiv.org/abs/math/0703216v1

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Sep 18, 2013
09/13

by
Christoph Lamm; Daniel Obermeyer

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We define cylinder knots as billiard knots in a cylinder. We present a necessary condition for cylinder knots: after dividing cylinder knots by possible rotational symmetries we obtain ribbon knots. We obtain an upper bound for the number of cylinder knots with two fixed parameters (out of three). In addition we prove that rosette knots are cylinder knots.

Source: http://arxiv.org/abs/math/9811006v1

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Sep 18, 2013
09/13

by
Stefan Friedl; Peter Teichner

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In the early 1980's Mike Freedman showed that all knots with trivial Alexander polynomial are topologically slice (with fundamental group Z). This paper contains the first new examples of topologically slice knots. In fact, we give a sufficient homological condition under which a knot is slice with fundamental group Z semi-direct product Z[1/2]. These two fundamental groups are known to be the only solvable ribbon groups. Our homological condition implies that the Alexander polynomial equals...

Source: http://arxiv.org/abs/math/0505233v3

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Jul 20, 2013
07/13

by
Neil R. Nicholson

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Although most knots are nonalternating, modern research in knot theory seems to focus on alternating knots. We consider here nonalternating knots and their properties. Specifically, we show certain classes of knots have nontrivial Jones polynomials.

Source: http://arxiv.org/abs/math/0609584v2

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Sep 20, 2013
09/13

by
V. Turaev

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Knots and links are interpreted as homotopy classes of nanowords and nanophrases in an alphabet consisting of 4 letters. Similar results hold for curves on surfaces. We also discuss versions of the Jones link polynomial and the link quandles for nanophrases.

Source: http://arxiv.org/abs/math/0506390v1

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Sep 20, 2013
09/13

by
Yair Minsky; Yoav Moriah; Saul Schleimer

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We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.

Source: http://arxiv.org/abs/math/0607265v3

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Sep 19, 2013
09/13

by
Vladimir Turaev

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We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.

Source: http://arxiv.org/abs/math/0703055v1

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Sep 22, 2013
09/13

by
Jenelle Marie McAtee

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In this paper we show how to realize all knot (and link) types as C^{2} smooth curves of constant curvature. Our proof is constructive: we build the knots with copies of a fixed finite number of "building blocks" that are particular segments of helices and circles. We use these building blocks to construct all closed braids.

Source: http://arxiv.org/abs/math/0403089v1

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Sep 21, 2013
09/13

by
Sebastian Baader

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We investigate the behaviour of Rasmussen's invariant $s$ under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

Source: http://arxiv.org/abs/math/0602092v2

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Jul 20, 2013
07/13

by
Peter Schmitt

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This note describes how to construct toroidal polyhedra which are homotopic to a given type of knot and which admit an isohedral tiling of 3-space.

Source: http://arxiv.org/abs/math/0107016v1

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Jul 20, 2013
07/13

by
Joan S. Birman; Nancy C. Wrinkle

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Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally simple} if it is determined by its topological knot type $\cK$ and its Bennequin number. The main theorem asserts that any $\cTK$ whose associated $\cK$ satisfies a condition...

Source: http://arxiv.org/abs/math/9910170v4

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Sep 19, 2013
09/13

by
Mario Eudave-Munoz

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We give a description of all (1,2)-knots in S^3 which admit a closed meridionally incompressible surface of genus 2 in their complement. That is, we give several constructions of (1,2)-knots having a meridionally incompressible surface of genus 2, and then show that any such surface for a (1,2)-knot must come from one of the constructions. As an application, we show explicit examples of tunnel number one knots which are not (1,2)-knots.

Source: http://arxiv.org/abs/math/0703132v3

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Sep 23, 2013
09/13

by
E. Denne

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It is known that for every knotted curve in space, there is a line intersecting it in four places, a quadrisecant. Comparing the order of the four points along the line and knot we can distinguish three types of quadrisecants; the alternating ones have the most relevance for the geometry of a knot. In this paper we prove that every (nontrivial tame) knot has an alternating quadrisecant. This result had applications to the total curvature, second hull and ropelength of knots.

Source: http://arxiv.org/abs/math/0510561v1

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Jul 20, 2013
07/13

by
Mathieu Baillif; David Cimasoni

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We study continuous embeddings of the long line L into L^n (n>1) up to ambient isotopy of L^n. We define the direction of an embedding and show that it is (almost) a complete invariant in the case n=2 for continuous embeddings, and in the case n>3 for differentiable ones. Finally, we prove that the classification of smooth embeddings L \to L^3 is equivalent to the classification of classical oriented knots.

Source: http://arxiv.org/abs/math/0406179v1

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Sep 20, 2013
09/13

by
Brooke Brennan; Thomas W. Mattman; Roberto Raya; Dan Tating

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Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realised by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q+1,q) torus knot is (2q+1)cot(\pi/(2q+1)) (resp., 2q cot(\pi/(2q+1))). Using these calculations, we provide the bounds c_1 \leq...

Source: http://arxiv.org/abs/math/0410565v1

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Sep 18, 2013
09/13

by
Liam Watson

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We give a recipe for constructing families of distinct knots that have identical Khovanov homology and give examples of pairs of prime knots, as well as infinite families, with this property.

Source: http://arxiv.org/abs/math/0606630v3

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Sep 20, 2013
09/13

by
Kazuhiro Ichihara; Shigeru Mizushima

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The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable subsurfaces bounded by the knot. In this paper, we determine the crosscap numbers of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos knots developed by Hatcher and Oertel.

Source: http://arxiv.org/abs/math/0608480v2

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Sep 17, 2013
09/13

by
Charles Livingston

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For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that if the metabolizer has a basis represented by a strongly slice link then K is slice. The question has been asked as to whether it is sufficient that each basis element is represented by a slice knot to assure that K is slice. For genus one knots this is of...

Source: http://arxiv.org/abs/math/0007088v1

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Sep 18, 2013
09/13

by
Jacob Mostovoy; Theodore Stanford

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We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse knots with one maximum that distinguishes two different representations of the figure eight knot. We also present the results of computer calculations for some invariants of low order. It turns out that for Morse knots with two maxima there is a Z/2-valued...

Source: http://arxiv.org/abs/math/0008096v1

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Sep 20, 2013
09/13

by
Mario Eudave-Munoz; Enrique Ramirez-Losada

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We determine all (1,1)-knots which admit an essential meridional surface, namely, we give a construction which produces (1,1)-knots having essential meridional surfaces, and show that if a (1,1)-knot admits an essential meridional surface then it comes from the given construction.

Source: http://arxiv.org/abs/math/0608205v1

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Sep 19, 2013
09/13

by
Tobias Ekholm; Maxime Wolff

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A holonomic knot is a knot in 3-space which arises as the 2-jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to the 1-jet extension of the function. There are two classical invariants of framed knot diagrams: the Whitney index (rotation number) W and the self linking number S. For a framed holonomic knot we show that W is bounded above by the negative of the braid...

Source: http://arxiv.org/abs/math/0206190v1

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Sep 22, 2013
09/13

by
Sze Kui Ng

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In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are one-to-one assigned with an integer such that prime knots are bijectively assigned with prime numbers and the prime number 2 corresponds to the trefoil knot. Then by considering the quantum knots as periodic orbits of the quantum system and by the identity of knots with integers and an approach...

Source: http://arxiv.org/abs/math/0603275v2

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Sep 22, 2013
09/13

by
Ryan Budney

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This paper gives a partial description of the homotopy type of K, the space of long knots in 3-dimensional Euclidean space. The primary result is the construction of a homotopy equivalence between K and the free little 2-cubes object over the space of prime knots. In proving the freeness result, a close correspondence is discovered between the Jaco-Shalen-Johannson decomposition of knot complements and the little cubes action on K. Beyond studying long knots in 3-space, we show that for any...

Source: http://arxiv.org/abs/math/0309427v6

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Sep 21, 2013
09/13

by
Abhijit Champanerkar; Ilya Kofman; Eric Patterson

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We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials.

Source: http://arxiv.org/abs/math/0311380v4

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Sep 18, 2013
09/13

by
Baptiste Chantraine

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In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding...

Source: http://arxiv.org/abs/math/0611848v3

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Sep 22, 2013
09/13

by
Gyo Taek Jin

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An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast to the fact that the difference between the sum of bridge indices of two knots and the bridge index of their connected sum is always one, the corresponding difference for the...

Source: http://arxiv.org/abs/math/0001090v2

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Sep 20, 2013
09/13

by
Eduardo Pina

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The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the stable and unstable manifolds, connecting the saddles. Each facxe is then oriented in one of two different senses determined by the direction of these manifolds. The associated matrix to that connected graph is decomposed in the sum of two permutations. The...

Source: http://arxiv.org/abs/math/0601199v1

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Sep 21, 2013
09/13

by
Vladimir Chernov

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It is well-known that a knot in a contact manifold $(M,C)$ transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class $e_C\in H^2(M)$ of $C$ is nonzero, then $C$ is nontrvivializable and the natural framing of transverse knots does not exist. We construct a new framing-type invariant of transverse knots called relative framing. It is defined for all tight $C$, all closed irreducible atoroidal...

Source: http://arxiv.org/abs/math/0104009v3

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Sep 18, 2013
09/13

by
Huseyin Aydin; Inci Gultekyn; Michele Mulazzani

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We obtain an explicit representation, as Dunwoody manifolds, of all cyclic branched coverings of torus knots of type $(p,mp\pm 1)$, with $p>1$ and $m>0$.

Source: http://arxiv.org/abs/math/0306439v1

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Sep 20, 2013
09/13

by
Daniel Moskovich

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For p=3 and for p=5 we prove that there are exactly p equivalence classes of p-coloured knots modulo (+/-1)--framed surgeries along unknots in the kernel of a p-colouring. These equivalence classes are represented by connect-sums of n left-hand (p,2)-torus knots with a given colouring when n=1,2,...,p. This gives a 3-colour and a 5-colour analogue of the surgery presentation of a knot.

Source: http://arxiv.org/abs/math/0506541v6

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Sep 17, 2013
09/13

by
Masakazu Teragaito

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The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surface bounded by the knot. We determine the crosscap numbers of torus knots.

Source: http://arxiv.org/abs/math/0207203v2